Lukewarm water in all sinks and showers

I am having a problem where all of my faucets and showers only release lukewarm water (about 91 degrees F). I have recently purchased this house and replaced the hot water heater. Originally we thought the problem was with the hot water heater. It was found that the new unit was defective (no dip tube). Today the unit was replaced again. This one is functioning and water is hot at the tank. With the new heater installed, For example, At the kitchen sink it will come out hot (120 degrees) for a few seconds then level off back to about 91. I am trying to get an idea of what the problem could be before I hire them to come back out and fix. I can provided further details if needed but don't know where to start.

I have found that if I run a faucet in any bathroom or shower, the kitchen faucet maintains hot water, where other running faucet stays warm. All the bathrooms are on one side of the house and the kitchen is on a seperate run on the opposite side of the house. To me its seems like cold water it somehow mixing with the hot water. By running both faucets it is pulling all the cold toward the bathroom lines allowing the kitchen to maintain temperature.

Sounds like a very possible hot/cold cross over.
Have you got a recirculation system?
Single handle faucets (moen or, mixet)?
Any possible connection where hot and cold can be turned on together with the outlet turned off?
Just installed a new Delta R-10000 tub/shower valve but haven't put the cartridge in yet?

Looks like all of my faucets have bad cartridges in them. I replaced 1 faucet already and am now getting better hot water from that one. I am going to replace the rest and the cartridges in the showers. Hopefully that resolves my problems.

Thanks for the help. I will post my findings once my work is complete.

I seen that 100's of times, Moen cartridges causing the hot and cold water to mix causing tempered water. Go through the building and replace all the cartridges and the people have good hot water. Make sure you set you water heaters thermostat to a safe operating temperature, Gama recommend it be set at 120, and most shower valves have a temp limit stop, after you set the temp to your water heater to your desired temp, you need to set the limit stops to give you no more than 110 degree water.

There are 5 bathroom sinks and 2 showers with single handle faucets.
Prior the the tank being replaced hot water was working fine in all rooms.

Nothing has changed but the hot water tank. I plan on replacing all the faucets but don't want to replace anything until the hot water issue gets straightened out.

Did junk from changing the water heater clog up all these faucets or what ? Otherwise the above statements doesn't make sense to me .

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I bet it was an unnoticed problem. The old heater was primely cranked up to max, thinking its old and needs to be that high to get some sort of hot water. And with the new unit the temp is set to the little arrows which is 120, now the bad cartridges are noticeable.